How to animate shape keys in Blend4Web
The way to animate Shape Keys in Blend4Web is to use Logic Nodes. It has been implied that in the future the Blend4Web team may add even more ways to animate them. (An alternative approach is to do vertex baking on models that have Shape Key animation on them.) I will give an overview of the steps for animating Shape Keys with Blend4Web's Logic Nodes.

Things you've done correctly already:


Set the render engine to Blend4Web
Create shape keys
In the Object Context, enable "Export Shape Keys" for that object.


Things you still need to do:


In the Scene Context enable the Logic Editor. Click the [+] button to add a new one.
Open the Node Editor and set the node tree type to Blend4Web logic nodes.
Select your new logic node tree from the drop-down list.
Create the kind of logic node tree necessary for the kind of animation you want to make.


Enabling the Logic Editor and creating a new node tree:



Setting the node tree type and selecting your logic node tree in the Node Editor:



Creating your node tree

Here is a very basic example of how you can make your object switch Shape Keys when clicked:



For examples of how to make your Shape Keys animate gradually over time, as well as how to animate back and forth, see this thread on the Blend4Web Forums.

Also, you can reference the Blend4Web Manual for more info on Logic Nodes.

Good luck!